I Love When They're Wrong

I love when they're wrong.  It's a beautiful moment if it happens with the right student-teacher relationship.

Allow me to explain my thinking here.  First of all, when I say "the right student-teacher relationship," I'm talking about an open facilitator type of role for the teacher, and a thinker role for the student.  It is based on trust, past failures and successes, and comes in time.  This is necessary because the student needs to know and understand that there is structure to each role, and that each role must be played out; it must be a relationship that is genuinely based on mutual respect and understanding - any belittling, "let me comfort you when you fail" or treat you like a kid garbage can't be happening here.  Yes, I did say garbage.  Yes, I did say don't comfort them when they fail.  Instead, we must provide them with a sort of flashlight to get to the end of the cave on their own, so that they know they can do it on their own, without our help the next time (or maybe the time after that, situation-depending).

Now that we've sort of covered what I think of the thinker-facilitator role, we'll get into the good stuff: being wrong.

I personally hate being wrong.  But then again, I was raised in a school culture of right and wrong; of failure and success; of this or that.  I hated math because in Gr. 8 I was told that I was wrong in front of the whole class, and then belittled with a primary example of addition (I believe the exact phrasing was "If we add 4 moo cows with 17 moo cows, how many moo cows are there?"  The numbers were probably different and who knows what we were learning, but there was definite referral to moo cows. In grade 8.  Really? Shut down.  It was a near-permanent shut down, until I started teaching.)  We need to allow our students to be wrong, to struggle through it uncomfortably, providing them with only the tools they absolutely require to succeed, and then have them come out successful.  This looks different for every child.

Some of my students require prompts.  For example, "Go and look at the "area" word card to remind you of what it is," or, "What's area again?" or as much as a mini lesson to reteach the basic concept.  I'm all about inquiry and exploratory learning, but if a child doesn't know how to speak or write, how can they tell their great tale?  If a child doesn't know yet what something like area is, and how it works, and how it could be found, how can they investigate it?  It's like saying, "I need you to bake me bread - welcome to my kitchen with no ingredients.  You can't leave, but get baking."  And the person only speaks Spanish.  And you're saying it in your head, so they don't even know what you're thinking, because they're not psychic and can't read your thoughts.  Hmmmm.

So once you've given them exactly and only what they require, you need to let them stew.  Let them work it out.  Some kids are visual - they'll need to draw it out, maybe on your chalk table.  Some kids are auditory - they'll need to talk it out with you.  Listen, but don't say much.  Stop encouraging them, for goodness sake.  Let them have a minute to think.  How annoying is it to have someone talking to you every time you're on the verge of a breakthrough?

I love when they're wrong because it means that they have a starting point.  It means there's something investigate.  We can work with a wrong answer, but no answer means there's nothing there at all to play with.  When they're wrong, they must struggle, and when they struggle and come out on top it means that they've made a memorable, authentic connection to what is being learned, and will likely carry that successful moment with them as they go through other struggles and learning experiences (as facilitator it's your job to help get them there without driving them ... just give them a map, or maybe a subway token, but seriously, stop leading them, they'll never LEARN that way - they'll only perform for you in that moment).

I love when they're wrong; not because I'm a jerk, but because it means they're working.  My job is to get them thinking, to get them working, and to have them learn something.  I can guarantee you: when you're doing the talking, the thinking, the helping ... there's no learning happening, except your own.  It's their time now, not yours, so sit back and be quiet - and let the magic happen.  Let them be wrong! Let them eat cake! No, let them divide the cake, and then eat it!  But be quiet while they do it.

Thinking Big

Pencils & papers have their place, but sometimes you need to get outside of the page to get outside of the box.

One way that I try to activate thinking is through the option of using a different workspace and medium.  I painted the top of an old table that we had around the house in chalk paint. 

 

Now, the kids can work at the table and do their "scratch work" in a more fun, engaging manner.  They are able to easily fix errors, because it's a simple wipe and not an erase that turns into a mucky looking paper.  When finished, they either transfer the work or I take a picture and save it to their file on my iPhone.

 

This method helped two kids on the "edge" of a breakthrough really grasp the concepts we've been working on.  They use a table to organize their work, and it opened up their thinking.

Interactive Word Walls

I've struggled with Word Walls for some time now.

To be honest, I completely understand the pedagogy behind them and love the idea of them ... but they just don't always work for me.  I have a hard time keeping up with them, taking the time to build them in class, and just seem to spend so much time making them pretty but not enough time making them work.  I've been wanting a Math Word Wall since I started teaching, but have never gotten to it.

Until, that is, I realized what the problem with a Math Word Wall is.  In Math, we use a lot of technical terms.  I don't baby it down or fancy it up, because there's enough lingo for the kids to grasp as it is.  What happens when a child mixes up perimeter and area?  What if they know how the word dimension works in a sentence, but they really aren't sure what it means in a question?  Just because they can spell it, doesn't mean they know it.

This is where I've made some changes to this whole word wall system:

1) The words are decided on by the class.  For example, if I say, "Okay class, today's problem involves numbers increasing," and one kid says, "Hey, that's a good math word, let's put it up!" (that happened today, by the way), then I will.  It takes me a whole minute to pull out the paper (that is pre-cut and ready to go) and write the word.

2) The words are on the front, but the definition is on the back.  It's an interactive Word Wall.

3) They words can come down.  They hang with one of those circular clips that you often see in Gr. 1 rooms with sight words.

4) The students are in charge.  They decide when to go and take down the word as a reference.  It gets them up and moving, they can take the word with them if they need it, and it's about meaning versus spelling.  They put them back, on the correct hook, and we can rearrange them as needed.  The kids sometimes mention them during a lesson: "Wait - what's a dimension again?  Can we get the word from the Word Wall?"


This is one way that I am trying to increase their vocabulary and an enriched understanding of the language of math, rather than a surface understanding of a word that they'll temporarily remember, until the test.

Semantics

Semantics is one of the biggest "problems" with math.  I fear that in early primary classes, math is so focused on number sense that when students are asked to start problem solving, words make all of the hard number sense work disappear.  It's as if it never happened!  The solution to this sounds simple (use more language in math at all grade levels) but isn't.

For the past couple of weeks, my students have been looking at the differences and similarities between perimeter and area.  I thought that they totally understood it, but wanted to do an assessment to check.  Thank goodness I did, because something about the phrasing seemed to cause them to forget everything.  AH!  They know what perimeter is, in isolation.  They know what area is, in isolation.  They can find different possible dimensions for a given perimeter, and can find all possible whole-number dimensions for a given area.  But, start to mix it up, and they're lost.

Slowly, as we toil away at different problems, it becomes more clear.  Here is where the importance of exposing students to different semantics comes in, as well as the importance of taking time.  Whereas we used to (well, I used to) drill the students with text book questions over and over and over, now the shift must be that this practice time is transformed in a way that allows the students to work over and over on something that is meaningful; something that is challenging.

Today I got to see the kids work with partners (that they chose, and boy, was I ever impressed in the choices that they made ... pairings I never thought would naturally happen, did!)  They were asked to solve one of these options, each offering a different challenge:

The area is 100 cm squared.  What could the perimeter be?

OR

The perimeter is 100 cm.  What could the dimensions be?

These questions are fairly (somewhat) straight forward.  The kids were comfortable with this.  When I gave them a map of my house, and asked them to pick a room, solve the area and perimeter, and then list other possible dimensions for the perimeter, it seemed to be too much.  The amount of words, even when broken up, were overwhelming.  The words area and perimeter suddenly became forgotten.  They couldn't quite figure out what it was that they needed to do. 

So my take-away questions, after a week of observing are:
- Do they just need more practice?
- Do they really understand the differences between perimter and area yet, or is it still muddy?
- How can I increase their skills and abilities from where they are now, to where I want them to be (able to use all of the skills and strategies we've been working on, by independently choosing the best option for them, whether it be a table, diagram, working backwards, or another option)?

The semantics in math can be overwhelming.  Changing the phrasing can terrify the kids and make them feel uncertain, even if they actually do know what to do.  Exposure to a variety of semantics is necessary, from an early age, with a consistency in what is being asked of them.  In this example, they need to be able to solve area, perimeter, and figure out different possible lengths and widths.  Solving area and perimeter should be kept consistent.  But, the depth of questions, the phrases and wording should be varied, and consolidated as a whole group.  Even in Grade 1! 

It takes time, and that time is bought with the use of layering in planning.  We're going back to some patterning next week, along with perimeter, by looking at a growing perimeter (I stole the question and modified it from a Math Makes Sense strategies toolkit question about a garden that gets bigger each year). 

Have you tried to change the phrasing in your questions?

Integration?

One of the problems with math is that it's done in isolation.  Math is so language based and so easily ties into science, social studies, arts and physical education, yet we so often set it aside, or make the integration of it a special event.  In the real world, it's not a special event - math is everywhere, like it or not.  We don't jump up and down when the cashier at Tim Horton's gives us the correct change, or when we're able to calculate the tip at a restaurant, or the taxes on a piece of clothing we REALLY want to buy, but might not have enough cash for.  So, why is math so isolated at school?

I don't have an answer.  All I can guess is that somewhere along the way, while training children to work in factories in the 1800's, it was determined that learning in isolation was the easiest way to train people.  Yes, it does work - look at all of us - we can do it.  But that doesn't mean it's fine, or the best option.  I think we need to constantly look for more opportunities for those "special events" and make math a regular part of every other subject, just as language is the base of every other subject.

I've been to workshops on integrating math in gym, and have seen perfect examples of how to do it with social studies (data management! year calculation!) ... it's so well blended into science, but we're still scared to make that transition out of saying that it's time for math, then science, then language, into saying that it's time to think and learn.  Instead of integrating learning, why aren't we simply learning?

I'm still experimenting, and will be for a few years, but my goal is to layer not just math expectations together - I want to layer different strands of math into every other subject, and to run learning blocks instead of subjects.  

Here's one way I am starting to experiment with this (let me preface it by saying that I'm still new to it, and it isn't yet a regular thing - but when I do run things like this, it isn't a special event; it's just time to learn):

Students investigate their sunflower.

I was cleaning up the yard last week when I got looking at the sunflowers.  As I cut them down, I realized I could save a ton of money on seeds next year by harvesting these.  So I grabbed a few of the sunflower "heads" and got to work.  I was amazed at the textures inside of the plant, but also at the patterns I was seeing.  It hit me that this would be a great way to wrap up our focus on patterning as a problem solving strategy, since we had already completed the assessment piece anyway.

This group pulled out a ruler to do some measuring.

I filled a bag with enough heads to share with the groups, and I asked the class to first make some observations.  It was a bit of a mind bender at first - you want us to do what? But I don't see anything!  But eventually, as they started to make diagrams and count, math was working its way into my science lesson.  I call it a lesson, but let's call a spade a spade a spade - I didn't teach a darn thing.  They did all of the work.  When I asked them trade off sunflowers ... now that's when it got really interesting.  This was the engagement moment.  They were hooked.  They started comparing things.  They started going back to the last flower to observe things they'd missed.  They started turning the flowers over to find patterns in the leaves, and they pulled out seeds to look at patterns making up the seeds.  They looked at the numbers of seeds.

Look at how engaged these boys are.  3 kids working with one flower?  Success!

Was this challenging, mathematically, for my grade five students?  Quite frankly, no.  But they were using the language of math, and using their eyes and minds to make observations.  The dialogue was rolling.  That was my goal: see what they can see.  It was challenging for them to look at something with very little direction, and to find some learning.  To that end, I believe it was worth it.

When I asked them to tear the sunflowers apart, and make observations about the insides - well that was just fun.  Educational fun, of course. ;)

We harvested all of the seeds to plant in the spring.  We'll be able to refer back to these moments now for more problems and problem-building (yes, I believe in challenging the kids to make their own problems, but we'll get to that later).  We'll also be able to talk about the experience when we talk about accountable group work.

That's one happy mathlete! 

This student's observations were more scientific than mathematical.  Is that okay?  Yes!

I know that you are reading this blog for math ideas and questions, and maybe this entry isn't as focused as you might have expected.  But, I think it's important to address the reality that math isn't always going to be focused - it needs to be messy and abstract as much as it needs to be focused and finite.  The experience creates the culture, and in exposing the kids to something from their own worlds (in this case, sunflowers from my garden) and letting them take an hour (yep, we spent an hour on it) to indulge into some serious thinking and observing, we're making math worth their time.  I rarely hear kids say that they hate math or suck at math anymore.  That's something worth thinking about.

One Question

One of the great things about teaching with Problem Based Math is that students are able to open up their work and extend their thinking, finding challenges to think harder about.  It is all about higher order thinking, which I do believe every child is capable of, within their zone of proximal development.

We had focused on using tables and looking for patterns, as well as following a five-step process for answering problems.  As an assessment piece, I wanted to see:

- can this child extend a pattern?
- can this child add or multiply (bonus if they multiply, because then they've extended their thinking into an algorithm)?
- can this child properly use a table and label each column in a way that makes sense?
- can this child prove their work?
- does this child give a final answer?

As an added bonus, I used measurements, to see a) if anyone converted when the numbers became large, and b) if they remembered to identify the unit of measurement.

The question itself was made up of manageable numbers.  I didn't want the challenge to be in the adding; I wanted it to be in reading the problem, understanding how to solve it, and showing their thinking/organizing their work.  For kids who were able to handle a challenge, or ones who needed a very easy-to-work-with question, I left the number of years wide open:

Draven was 50 cm tall when he was born.  If he grows 40 cm every year, how tall will he be when he is ___ years old?

You could also change the question by:
- changing the numbers for measurement, giving options, or leaving them (both or just one) blank
- asking how old he will be when he is a certain cm. (given, left blank, or options) tall


Have a look at how this student solved the problem.  It's nothing amazing, but rather average, until you get into the "meat and potatoes" - the PROOF.  This explanation is beyond awesome!

You can see that the student:
- solved the problem
- organized the answer into a table with appropriate headings
- chose an appropriate challenge of "14 years" (rather than 4, 8 or 10, for example)
- explained the answer
- explained how tall he was at birth, and how he knows not only based on the question, but also based on his work, by doing a reverse-operation and subtracting
- clearly answered the question

This student answered in a fantastic way.  This child works well above other students, who also completed the question successfully.

This is a video that shows a student working on the same question, but using cm cubes as counters.  On his own, he counted each block as 10 units to save time, and counted up the groups.  It's not ground breaking to watch, but beautiful to see a child finding their own way.  Later, he transferred the work into a table and answered the question.  I didn't tell him to do any of this, and although he learns differently from other students, he was still able to come out on top.  This is the beauty of open questions.

By getting our heads out of text books and into problems with open possibilities, the kids are able to think deeper, look farther and show everything they're capable of doing.

The Zone

The Zone of Proximal Development is huge in education.  Basically, you can't learn if you have nothing for the learning to cling to, or if it's so dang easy that you could do it in your sleep.  I'm finding that this year more than ever, I'm hitting the zone for my students, evidenced by:

- on task behaviour
- thoughtful questioning
- perseverance
- success of their answers

When conducting Problem Based Math, it is absolutely necessary to pick your curriculum expectations before starting the questions, and look at the "long run" - I look at about 4-5 days at a time.  I ask myself:

1) What strands can go together based on this "inspiration expectation"?
The inspiration expectation should be your MAIN expectation: this is what you'll base everything else around.

2) How many expectations are manageable with my main inspiration expectation?
When starting, only 2 may be manageable.  I find that the basic number sense expectations go pretty much with everything, so those remain most weeks, or cycles as I call them, as my five day plans don't always land inside a perfect week.

3) What strategy/strategies do I want the kids to focus on/learn/build with this cycle?
If we're going to focus on building a table, then I know my expectations should fit into a question that requires a table.  This week, I'm focusing on finding different dimensions for a specific area and/or perimeter, so a table may come in handy, but we'll layer in Working Backwards, so that the students can focus on the relationship between multiplying and dividing.  (Example: The area is 36cm.  Since 36 divided by 9 is 4, I know that the dimensions of the rectangle are 9cm (length) and 4cm (width).  9x4=36 and 36 divided by 9 is 4.  I can work backwards from the answer of a=36cm.)

When I know this, I look for expectations from each strand that could fit nicely together.  I can usually layer in about 3 strands comfortably.

Then, I take the prompts directly from the curriculum to get started.  The example in #3 is from a prompt.  That prompt will be the focus of our learning for 3 out of 5 days of this cycle, worded in different ways, so that an understanding of area, perimeter, operations, organization, and patterning/algebra (missing numbers in an equation) is built.

I start day 1 of each cycle by introducing or revisiting a problem solving strategy.  The questions I ask are easy, but the students are regularly reminded that the numbers are easy so that they can focus on understanding the strategy.  This is how I access their zone for the strategy building part of the week.  Once that strategy has been practiced as a whole group, with peers, and independently on day 1 (each problem, which comes from different problem solving resources, including Problem Solver and Word Problems for The SmartBoard), we reflect on the strategy in our math journals, and call it a day.


As we progress through days 2-4, their zones are established by reminding them of the strategies they can use, the problem solving steps they can use, and by reading the problem together as a class.  Really, the kids do most of the work.  I rarely say anything other than their names so they can "teach."

During the problem solving period (about 30-40 minutes) on days 2-4, I am constantly circulating, to see where the kids are and how they're progressing.  Some need prompts, but I try to keep those rare, only when they're absolutely necessary.  This is how I stay on top of who is in the zone and who is lost.

Once I've figured that out, I can pull small groups for "Guided Math," and at the end of each problem solving period, the kids volunteer to show off their work.  I'm lucky enough to have a document camera with my Smart Board, so we can show their work on "the big screen."  The kids volunteer even when they know they've made a mistake, because they have something show off: maybe it's their organizing skills, or maybe it's their proof statement.  Regardless, they have something to share, so I encourage it.  It builds confidence and exposes students to possible errors, different thinking, and different correct answers.

On Day 5, I do a "check."  This is an independent assessment question that has nothing new to it.  It isn't a copy of the others, but it uses similar wording, numbers and expects the same from the students.  Otherwise, it would be an unfair assessment and a waste of time.

By constantly monitoring and going into curriculum documents on a weekly (sometimes daily!) basis, you'll hit their zones no problem.

The curriculum documents fully support problem based math, and are one of the few resources you really need.  Imagine clearing the bookshelves of old and tired copiable worksheets and text books, and placing manipulatives on the shelves instead.  It's possible.  I know, because I'm doing it!

The Problem With Math

The problem with math?

There are so many I don't know if I can count them.  (Is that a problem right there?)

With so many old and new (and good and not-so-good) ideas floating around, it becomes confusing about what and how we should be teaching math.  We tend to rely on the structure from text books, which, unfortunately, causes us to sometimes become a bit too relaxed with planning and maybe (definitely) sucks the life right out of what can be a fun and interactive subject.  It tends to close up our communication and even when we often THINK we're being open and clear, we're often shutting down (new ways of) thinking.  This is what happened to me - I wasn't right, so I was told so, so I shut down.

I've been using the Marion Small approach to math, called Problem Based Math, for two years (and one month) now, and can assure you that it makes perfect sense!  I don't teach using one strand (Geometry, or Measurement, or Patterning, etc...) at a time, but often tie in 2 or 3 (sometimes all 5!) into a lesson.  There are no units.  There are no major "TESTS" - just check ins and assessments so that I can see how well a child can solve a problem, how their fundamental skills are, and what I can do better as a facilitator.  I choose a focus per week so that I know they are learning (we focused on using tables the past three weeks - can they ever rock a table now, even though we worked on patterning, measuring, adding, multiplying, geometry and data management).  We build on prior knowledge and work on one question a day.  We present the ideas and work, even when it's wrong, and feel good about it.

Students measure the perimeter of a floor tile, and then multiply it by the number of their choice,
to find out how many times it needs to be multiplied to reach 300cm.

It's about a culture of thinking, not a culture of what I've taught or what they've "learned" (memorized? learned for now?).  It's about showing and speaking thinking instead of keeping it bottled up inside.  It's about making it real, making it meaningful and making it count.

There are no new ideas here.  They're borrowed from the actual curriculum documents we're meant to teach by (seriously, most of the work is already done, and those documents didn't cost me a cent), from the research conducted by experts (Marion Small and the ever-extremist Van DeWalle - whose ideas I've boiled down to fit the timing of the school day); the ideas are built on what the kids know and how they work; from games and activities in the real world.  The way I teach math involves little teaching - but what is instructed is necessary and meaningful.  

From a kid who hated math, to an adult who teaches it, I've learned a lot more in the past three years than I ever did with a text book or a lecturer (as a teacher and as a kid).  I'll be using this blog to post lesson ideas and experiences - how I've come to teach math this way, and what you might want to think about to change your own practice.  I hope you'll stick with me so that you can see that the "problem" with math should be the math problem, and nothing else.